The measurement problem is one of the most relevant open problems of quantum mechanics. What is a measurement? What constitutes an observer and what doesn't? Is the wavefunction a physical object (ontological) or just a mathematical construct that represents our ignorance of the state of a system?
Trying to answer these questions has produced a multitude of interpretations of quantum mechanics.
The Copenhagen interpretation is the most famous. It basically states that small things are quantum, big things are classical, and when a small thing interacts with a big thing there is a measurement and a collapse of the wave function for everyone, while when small things interact with each other the just entangle and evolve unitarily, without any collapse. The problem with this interpretation is that it doesn't say where we should draw the line between big and small.
The Many World interpretations on the other hand treats everything (big and small things) as a quantum system. Everything evolves unitarily, there is never any collapse. The drawback is that this means that you are a quantum system, and so you can be in a superposition. And this is difficult to reconcile with the idea of a "self".
Other interpretations, like quantum Bayesianism advocate that a quantum state is just a representation of the degree of belief a person may have about the outcome of some measurements. The wavefunction is therefore subjective and not an objective physical object. According to quantum Bayesianism, the collapse of the wavefunction is just a Bayesian update of the beliefs of the observer. In this sense, a wavefunction may collapse for me and not for you.
As you can see, the question of the collapse of the wavefunction is far from settled. After a measurement it may collapse for everyone, just for someone or it may never collapse at all. If this seems like speculation to you, and not scientific facts, you are right. As of today, nobody has been able to experimentally distinguish between these (and others) interpretations.
What Frauchiger and Renner are trying to do in their famous paper is to devise a thought experiment that may be able to rule out some of these interpretations, by showing a contradiction between a set of well accepted assumptions.